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Nonconvex Separation Theorems

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Set-valued Optimization

Part of the book series: Vector Optimization ((VECTOROPT))

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Abstract

In this chapter we introduce nonlinear scalarization methods that are very important from the theoretical as well as computational point of view. We introduce different scalarizing functionals and discuss their properties, especially monotonicity, continuity, Lipschitz continuity, sublinearity, convexity. Using these nonlinear functionals we show nonconvex separation theorems. These nonlinear functionals are used for deriving necessary optimality conditions for solutions of set-valued optimization problems in Sect. 12.8 and in different proofs, especially in the proof minimal point theorems in Chap. 10. Moreover, we study characterizations of solutions of set-valued optimization problems by means of nonlinear scalarizing functionals. Finally, we present the extremal principle by Kruger and Mordukhovich and discuss its relationship to separation properties of nonconvex sets. This extremal principle will be applied in Sect. 12.9 for deriving a subdifferential variational principle for set-valued mappings and in Sect. 12.11 in order to prove a first order necessary condition for fully localized minimizers of set-valued optimization problems.

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References

  1. Artzner, P., Delbaen, F., Eber, J.M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122(2, Ser. A), 301–347 (2010)

    Google Scholar 

  3. Bao, T.Q., Mordukhovich, B.S.: Set-valued optimization in welfare economics. Adv. Math. Econ. 13, 113–153 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bao, T.Q., Tammer, C.: Lagrange necessary conditions for Pareto minimizers in Asplund spaces and applications. Nonlinear Anal. 75(3), 1089–1103 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonnisseau, J.M., Cornet, B.: Existence of equilibria when firms follow bounded losses pricing rules. J. Math. Econ. 17, 119–147 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonnisseau, J.M., Crettez, B.: On the characterization of efficient production vectors. Econ. Theory 31, 213–223 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Borwein, J.M., Zhu, Q.J.: Techniques of variational analysis. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 20. Springer, New York (2005)

    Google Scholar 

  8. Durea, M., Tammer, C.: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449–467 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch. 4, 429–447 (2002)

    Article  Google Scholar 

  10. Föllmer, H., Schied, A.: Stochastic Finance. Walter de Gruyter, Berlin (2004)

    Book  MATH  Google Scholar 

  11. Gerstewitz(Tammer), C.: Nichtkonvexe Dualität in der Vektoroptimierung. Wissenschaftliche Zeitschrift der TH Leuna-Merseburg 25(3), 357–364 (1983)

    Google Scholar 

  12. Gerstewitz(Tammer), C., Iwanow, E.: Dualität für nichtkonvexe Vektoroptimierungsprobleme. Wiss. Z. Tech. Hochsch. Ilmenau 2, 61–81 (1985)

    Google Scholar 

  13. Gerth, C., Weidner, P.: Nonconvex separation theorems and some applications in vector optimization. J. Optim. Theory Appl. 67(2), 297–320 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational methods in partially ordered spaces. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 17. Springer, New York (2003)

    Google Scholar 

  15. Göpfert, A., Tammer, C., Zălinescu, C.: On the vectorial Ekeland’s variational principle and minimal points in product spaces. Nonlinear Anal. 39(7, Ser. A: Theory Methods), 909–922 (2000)

    Google Scholar 

  16. Gorokhovik, V.V, Gorokhovik, S.Ya.: A criterion of the global epi-Lipschitz property of sets (Russian). Izvestiya Natsional’no Akademii Nauk Belarusi. Seriya Fiziko-Matematicheskikh Nauk, pp. 118–120 (1995)

    Google Scholar 

  17. Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Nonlinear Analysis and Variational Problems, Springer Optim. Appl., vol. 35, pp. 305–324. Springer, New York (2010)

    Google Scholar 

  18. Hamel, A.H.: Equivalents to Ekeland’s variational principle in uniform spaces. Nonlinear Anal. 62(5), 913–924 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hernández, E., Rodríguez-Marín, L.: Nonconvex scalarization in set optimization with set-valued maps. J. Math. Anal. Appl. 325(1), 1–18 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Heyde, F.: Coherent risk measures and vector optimization. In: Küfer, K.H., Rommelfanger, H., Tammer, C., Winkler, K. (eds.) Multicriteria Decision Making and Fuzzy Systems, pp. 3–12. SHAKER, Aachen (2006)

    Google Scholar 

  21. Hiriart-Urruty, J.B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4(1), 79–97 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ioffe, A.D.: Variational analysis and mathematical economics 1: Subdifferential calculus and the second theorem of welfare economics. In: Advances in Mathematical Economics, Vol. 12. Adv. Math. Econ., vol. 12, pp. 71–95. Springer, Tokyo (2009)

    Google Scholar 

  23. Krasnosel′skiĭ, M.A.: Positive solutions of operator equations. Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron. P. Noordhoff Ltd. Groningen (1964)

    Google Scholar 

  24. Kruger, A.J., Morduhovič, B.Š.: Extremal points and the Euler equation in nonsmooth optimization problems. Dokl. Akad. Nauk BSSR 24(8), 684–687, 763 (1980)

    Google Scholar 

  25. Kruger, A.Y.: On the extremality of set systems. Dokl. Nats. Akad. Nauk Belarusi 42(1), 24–28, 123 (1998)

    Google Scholar 

  26. Kuroiwa, D.: Existence Theorems of Set Optimization with Set-Valued Maps. Manuscript Shimane University, Japan (1997)

    Google Scholar 

  27. Luc, D.T.: Theory of vector optimization. Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)

    Google Scholar 

  28. Luenberger, D.G.: New optimality principles for economic efficiency and equilibrium. J. Optim. Theory Appl. 75(2), 221–264 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mordukhovich, B.S.: Maximum principle in the problem of time optimal response with nonsmooth constraints. Prikl. Mat. Meh. 40(6), 1014–1023 (1976)

    MathSciNet  Google Scholar 

  30. Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183(1), 250–288 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications. Springer, Berlin (2006)

    Google Scholar 

  32. Mordukhovich, B.S.: Variational analysis and generalized differentiation, Vol. II: Applications, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 331. Springer, Berlin (2006)

    Google Scholar 

  33. Pascoletti, A., Serafini, P.: Scalarizing vector optimization problems. J. Opt. Theory Appl. 42, 499–524 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rocca, M.: Well-posed vector optimization problems and vector variational inequalities. J. Inf. Optim. Sci. 27(2), 259–270 (2006)

    MathSciNet  MATH  Google Scholar 

  35. Rockafellar, R.T.: Clarke’s tangent cones and the boundaries of closed sets in \(\mathbb{R}^{n}\). Nonlinear Anal. Theory Methods Appl. 3, 145–154 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rockafellar, R.T.: The Theory of Subgradients and Its Applications to Problems of Optimization, R & E, vol. 1. Heldermann Verlag, Berlin (1981)

    Google Scholar 

  37. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000)

    Google Scholar 

  38. Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26, 1443–1471 (2002)

    Article  Google Scholar 

  39. Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Deviation measures in risk analysis and optimization. Preprint 7 (2002)

    Google Scholar 

  40. Rubinov, A., Singer, I.: Topical and sub-topical functions. downward sets and abstract convexity. Optimization 50, 307–351 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rubinov, A.M.: Sublinear operators and their applications. Uspehi Mat. Nauk 32(4(196)), 113–174, 287 (1977)

    Google Scholar 

  42. Ruszczyński, A., Shapiro, A.: Optimization of convex risk functions. Math. Oper. Res. 31(3), 433–452 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tammer, C., Zălinescu, C.: Vector variational principles for set-valued functions. In: Recent Developments in Vector Optimization, pp. 367–415. Springer, Berlin (2012)

    Google Scholar 

  44. Tammer, C., Zălinescu, C.: Lipschitz properties of the scalarization function and applications. Optimization 59, 305–319 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Rochester Institute of Technology School of Mathematical Sciences, Rochester, New York, USA

    Akhtar A. Khan

  2. Halle, Germany

    Christiane Tammer

  3. Faculty of Mathematics, University “Al. I. Cuza” Iasi, Iasi, Romania

    Constantin Zălinescu

Authors
  1. Akhtar A. Khan
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  2. Christiane Tammer
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  3. Constantin Zălinescu
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Khan, A.A., Tammer, C., Zălinescu, C. (2015). Nonconvex Separation Theorems. In: Set-valued Optimization. Vector Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54265-7_5

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